The object of the present paper is to introduce a type of non-flat Riemannian manifolds called almost pseudo cyclic Ricci symmetric manifold and study its geometric properties. Among others it is shown that an almost pseudo cyclic Ricci symmetric manifold is a special type of quasi-Einstein manifold. We also study conformally flat almost pseudo cyclic Ricci symmetric manifolds and prove that such a manifold is isometrically immersed in an Euclidean manifold as a hypersurface. The existence of such notion is ensured by a non-trivial example.

In this paper we prove that a spectraloid operator is finite, we present some generalized finite operators and we give a new class of finite operators. Also, the orthogonality of some operators is studied.

The symmetric rank-one (SR1) method is one of the well-known quasi-Newton methods, and many researchers have studied the SR1 method. On the other hand, to accelerate quasi-Newton methods, some researchers have proposed variants of the secant condition. In this paper, we propose SR1 methods based on some modified secant conditions. We analyze local behaviors of the methods. In order to establish the global convergence of the methods, we apply the trust region method to our methods.

We extend a theorem of Tits about the fundamental groups of graphs of Coxeter groups to those of braid monoids. More precisely, we show that every self-homotopy of a word decomposes into self-homotopies each of which is inessential, a cube, a prism or a permutohedron.

In this paper, we propose a new definition for multivariate kurtosis based on the two measures of multivariate kurtosis defined by Mardia (1970) and Srivastava (1984), respectively. Under normality, the expectation and the variance for the new multivariate sample measure of kurtosis are given exactly. We also give the third moment for the sample measure of new multivariate kurtosis. After that standardized statistics and normalizing transformation statistic for the sample measure of a new multivariate kurtosis are derived by using these results. Finally, in order to evaluate accuracy of these statistics, we present the numerical results by Monte Carlo simulation for some selected values of parameters.

In this paper, we propose a new pseudorandom number generator (AST) using an Artin-Schreier tower, which is a modified version of the twisted generalized feedback shift register (TGFSR). In TGFSR, the period is depending on the order of its multiplied matrix, and it is difficult to get the theoretical upper bound in general. Using the recursive structure of Artin-Schreier towers, we define a matrix with a parameter whose order is expected to be close to fairly near the upper bound. After examining some properties of this matrix, we give an algorithm of a new pseudorandom number generator AST which produce a sequence of numbers with a long period. Finally, we report the results of TestU01 to qualify that it has a good statistical property, although its generation speed is rather slower than TGFSR.

Let $\mathcal{G}(n;{\theta }_{2k+1},\ge \delta )$ denote the class of non-bipartite ${\theta }_{2k+1}$-free graphs on $n$ vertices and minimum degree at least $\delta$ and let $f(n;{\theta }_{2k+1},\ge \delta )=\text{max}\{ℰ(G):G\in \mathcal{G}(n;{\theta }_{2k+1},\ge \delta )\}$. In this paper we determine an upper bound of $f(n;{\theta }_7,\ge 25)$ by proving that for large $n$, $f(n;{\theta }_7,\ge 25)\le \lfloor \frac{{(n-2)}^2}{4}\rfloor +3$. Our result confirm the conjecture made in [1], ”Some extermal problems in graph theory”, Ph.D thesis, Curtin University of Technology, Australia (2007), in case $k=3$ and $\delta =25$.